In

category theory
Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled directed edges are cal ...

, an abstract mathematical discipline, a nodal decomposition of a morphism $\backslash varphi:X\backslash to\; Y$ is a representation of $\backslash varphi$ as a product $\backslash varphi=\backslash sigma\backslash circ\backslash beta\backslash circ\backslash pi$, where $\backslash pi$ is a strong epimorphism, $\backslash beta$ a bimorphism, and $\backslash sigma$ a strong monomorphism.A monomorphism
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In the context of abstract algebra or universal algebra, a monomorphism is an Injective function, injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y.
In the more general setting of catego ...

$\backslash mu:C\backslash to\; D$ is said to be strong, if for any epimorphism
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In category theory
Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labe ...

$\backslash varepsilon:A\backslash to\; B$ and for any morphisms $\backslash alpha:A\backslash to\; C$ and $\backslash beta:B\backslash to\; D$ such that $\backslash beta\backslash circ\backslash varepsilon=\backslash mu\backslash circ\backslash alpha$ there exists a morphism $\backslash delta:B\backslash to\; C$, such that $\backslash delta\backslash circ\backslash varepsilon=\backslash alpha$ and $\backslash mu\backslash circ\backslash delta=\backslash beta$
Uniqueness and notations

If exists, the nodal decomposition is unique up to an isomorphism in the following sense: for any two nodal decompositions $\backslash varphi=\backslash sigma\backslash circ\backslash beta\backslash circ\backslash pi$ and $\backslash varphi=\backslash sigma\text{'}\backslash circ\backslash beta\text{'}\backslash circ\backslash pi\text{'}$ there existisomorphism
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s $\backslash eta$ and $\backslash theta$ such that
: $\backslash pi\text{'}=\backslash eta\backslash circ\backslash pi,$
: $\backslash beta=\backslash theta\backslash circ\backslash beta\text{'}\backslash circ\backslash eta,$
: $\backslash sigma\text{'}=\backslash sigma\backslash circ\backslash theta.$
This property justifies some special notations for the elements of the nodal decomposition:
:$\backslash begin\; \&\; \backslash pi=\backslash operatorname\_\backslash infty\; \backslash varphi,\; \&\&\; P=\backslash operatorname\_\backslash infty\; \backslash varphi,\backslash \backslash \; \&\; \backslash beta=\backslash operatorname\_\backslash infty\; \backslash varphi,\; \&\&\; \backslash \backslash \; \&\; \backslash sigma=\backslash operatorname\_\backslash infty\; \backslash varphi,\; \&\&\; Q=\backslash operatorname\_\backslash infty\; \backslash varphi,\; \backslash end$
– here $\backslash operatorname\_\backslash infty\; \backslash varphi$ and $\backslash operatorname\_\backslash infty\; \backslash varphi$ are called the ''nodal coimage of $\backslash varphi$'', $\backslash operatorname\_\backslash infty\; \backslash varphi$ and $\backslash operatorname\_\backslash infty\; \backslash varphi$ the ''nodal image of $\backslash varphi$'', and $\backslash operatorname\_\backslash infty\; \backslash varphi$ the ''nodal reduced part of $\backslash varphi$''.
In these notations the nodal decomposition takes the form
:$\backslash varphi=\backslash operatorname\_\backslash infty\; \backslash varphi\backslash circ\backslash operatorname\_\backslash infty\; \backslash varphi\; \backslash circ\; \backslash operatorname\_\backslash infty\; \backslash varphi.$
Connection with the basic decomposition in pre-abelian categories

In a pre-abelian category $$ each morphism $\backslash varphi$ has a standard decomposition : $\backslash varphi=\backslash operatorname\; \backslash varphi\backslash circ\backslash operatorname\; \backslash varphi\backslash circ\backslash operatorname\; \backslash varphi$, called the ''basic decomposition'' (here $\backslash operatorname\; \backslash varphi=\backslash ker(\backslash operatorname\; \backslash varphi)$, $\backslash operatorname\; \backslash varphi=\backslash operatorname(\backslash ker\backslash varphi)$, and $\backslash operatorname\; \backslash varphi$ are respectively the image, the coimage and the reduced part of the morphism $\backslash varphi$). If a morphism $\backslash varphi$ in a pre-abelian category $$ has a nodal decomposition, then there exist morphisms $\backslash eta$ and $\backslash theta$ which (being not necessarily isomorphisms) connect the nodal decomposition with the basic decomposition by the following identities: : $\backslash operatorname\_\backslash infty\; \backslash varphi=\backslash eta\backslash circ\backslash operatorname\; \backslash varphi,$ : $\backslash operatorname\; \backslash varphi=\backslash theta\backslash circ\backslash operatorname\_\backslash infty\; \backslash varphi\backslash circ\backslash eta,$ : $\backslash operatorname\_\backslash infty\; \backslash varphi=\backslash operatorname\; \backslash varphi\backslash circ\backslash theta.$Categories with nodal decomposition

A category $$ is called a ''category with nodal decomposition'' if each morphism $\backslash varphi$ has a nodal decomposition in $$. This property plays an important role in constructing envelope (category theory), envelopes and refinement (category theory), refinements in $$. In an abelian category $$ the basic decomposition : $\backslash varphi=\backslash operatorname\; \backslash varphi\backslash circ\backslash operatorname\; \backslash varphi\backslash circ\backslash operatorname\; \backslash varphi$ is always nodal. As a corollary, ''all abelian categories have nodal decomposition''. ''If a pre-abelian category $$ is linearly complete, A category $$ is said to be ''linearly complete'', if any functor from a linearly ordered set into $$ has direct limit, direct and inverse limits. well-powered in strong monomorphismsA category $$ is said to be ''well-powered in strong monomorphisms'', if for each object $X$ the category $\backslash operatorname(X)$ of all strong monomorphisms into $X$ is skeletally small (i.e. has a skeleton which is a set). and co-well-powered in strong epimorphisms,A category $$ is said to be ''co-well-powered in strong epimorphisms'', if for each object $X$ the category $\backslash operatorname(X)$ of all strong epimorphisms from $X$ is skeletally small (i.e. has a skeleton which is a set). then $$ has nodal decomposition.'' More generally, ''suppose a category $$ is linearly complete, well-powered in strong monomorphisms, co-well-powered in strong epimorphisms, and in addition strong epimorphisms discern monomorphismsIt is said that ''strong epimorphisms discern monomorphisms'' in a category $$, if each morphism $\backslash mu$, which is not a monomorphism, can be represented as a composition $\backslash mu=\backslash mu\text{'}\backslash circ\backslash varepsilon$, where $\backslash varepsilon$ is a strong epimorphism which is not an isomorphism. in $$, and, dually, strong monomorphisms discern epimorphismsIt is said that ''strong monomorphisms discern epimorphisms'' in a category $$, if each morphism $\backslash varepsilon$, which is not an epimorphism, can be represented as a composition $\backslash varepsilon=\backslash mu\backslash circ\backslash varepsilon\text{'}$, where $\backslash mu$ is a strong monomorphism which is not an isomorphism. in $$, then $$ has nodal decomposition.'' The category Ste of stereotype spaces (being non-abelian) has nodal decomposition, as well as the (non-additive category, additive) category SteAlg of stereotype algebras .Notes

References

* * *{{cite journal, last=Akbarov, first=S.S., title=Envelopes and refinements in categories, with applications to functional analysis, url=https://www.impan.pl/en/publishing-house/journals-and-series/dissertationes-mathematicae/all/513, journal=Dissertationes Mathematicae, year=2016, volume=513, pages=1–188, arxiv=1110.2013, doi=10.4064/dm702-12-2015, s2cid=118895911 Category theory